Subscribe to this blog

Follow by Email

Search This Blog

The Sultan's Dowry Puzzle

Q: A sultan has a 100 daughters. Each of them has a fixed amount of dowry associated. Nothing is known about the distribution of dowry amounts among the princesses. You are to pick one of them. You are allowed to know the dowry amount following which you have to make a decision on whether to accept that princess or not. If you reject the princess you move on to the next princess and you cannot come back to the princess you have rejected. Your goal is to pick the princess with maximum amount of dowry. What should your optimal strategy be and what is your probability of success?

A: This one is a classic problem and there is adequate coverage around the web on how to solve this particular puzzle (which is also popularly called the secretary problem/interviewer problem etc). It is also a great introduction to one of the most interesting areas in probability (multi arm bandits/optimal stopping etc). The optimal solution has the form of sampling the first "r" princesses, taking note of the highest amount seen up until "r" and then subsequently picking an amount that is greater than this chosen maximum. To understand this, take a look at the following figure, each box being an amount, and let "n" be the number of princesses.

A first step to understand is the probability of the maximum value being on any index "i". This is simple. It is 1/n. Next, the strategy that we are adopting, would work only if the maximum amount we have seen so far (i.e. up until index i-1) lies in between the first index and "r". This happens with probability

Note, this is the probability of success from index "i". To get the total probability we sum up the above with "i" running from "r+1" to "n".

From this point on we solve this by either running a program in any computer programming language (R say) or we can make some approximations. We can let n run to infinity and replace the summation by an integral (transforming x = r/n) to yield

In order to maximize P(Success), we first differentiate it with respect to x and set it to zero (note: second derivate is negative so it must correspond to a maximum)
This yields the optimal value of x to be 1/e, which translates as

The maximum probability of success is also 1/e, which is a surprising lift this strategy provides, given that a random selection would have given a win probability of 1/n.

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

Linear Algebra (Dover Books on Mathematics)
An excellent book to own if you are looking to get into, or want to understand linear algebra. Please keep in mind that you need to have some basic mathematical background before you can use this book.

Linear Algebra Done Right (Undergraduate Texts in Mathematics)
A great book that exposes the method of proof as it used in Linear Algebra. This book is not for the beginner though. You do need some prior knowledge of the basics at least. It would be a good add-on to an existing course you are doing in Linear Algebra.

Follow @ProbabilityPuzIf you are looking to learn time series analysis, the following are some of the best books in time series analysis.

Introductory Time Series with R (Use R!)
This is good book to get one started on time series. A nice aspect of this book is that it has examples in R and some of the data is part of standard R packages which makes good introductory material for learning the R language too. That said this is not exactly a graduate level book, and some of the data links in the book may not be valid.

Econometrics
A great book if you are in an economics stream or want to get into it. The nice thing in the book is it tries to bring out a oneness in all the methods used. Econ majors need to be up-to speed on the grounding mathematics for time series analysis to use this book. Outside of those prerequisites, this is one of the best books on econometrics and time series analysis.